Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Lei Wen ◽  
Subin Zhuang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Order ◽  
Fractional Laplacian ◽  
Spatial Variable ◽  
Variable Order ◽  
New Wave ◽  
Dissipation Term ◽  
Laplacian Operators ◽  
Dispersion Term

We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.

Two efficient modeling schemes for fractional Laplacian viscoacoustic wave equation

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. T233-T249 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
Qingqing Li ◽  
Yufeng Wang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Computational Efficiency ◽  
Fractional Laplacian ◽  
Heterogeneous Media ◽  
Spatial Variable ◽  
Variable Order ◽  
Wide Range ◽  
Seismic Quality Factor ◽  
Efficient Modeling

Recently, a decoupled fractional Laplacian viscoacoustic wave equation has been developed based on the constant-[Formula: see text] model to describe wave propagation in heterogeneous media. We have developed two efficient modeling schemes to solve the decoupled fractional Laplacian viscoacoustic wave equation. Both schemes can cope with spatial variable-order fractional Laplacians conveniently, and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media. Both schemes are based on fast Fourier transform, and have a spectral accuracy in space. The first scheme solves a modified wave equation with constant-order fractional Laplacians instead of spatial variable-order fractional Laplacians. Due to separate discretization of space and time, the first scheme has only first-order accuracy in time. Differently, the second scheme is based on an analytical wave propagator, and has a higher accuracy in time. To increase computational efficiency of the second modeling scheme, we have adopted the low-rank decomposition in heterogeneous media. We also evaluated the feasibility of applying an empirical approximation to approximate the fractional Laplacian that controls amplitude loss during wave propagation. When the empirical approximation is applied, our two modeling schemes become more efficient. With the help of numerical examples, we have verified the accuracy of our two modeling schemes with and without applying the empirical approximation, for a wide range of seismic quality factor ([Formula: see text]). We also compared computational efficiency of our two modeling schemes using numerical tests.

A constant fractional-order viscoelastic wave equation and its numerical simulation scheme

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. T39-T48 ◽  
Author(s):  
Ning Wang ◽  
Hui Zhou ◽  
Hanming Chen ◽  
Muming Xia ◽  
Shucheng Wang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Seismic Wave Propagation ◽  
Variable Order ◽  
New Wave ◽  
Viscoelastic Wave Equation ◽  
Viscoelastic Wave ◽  
Order Scheme ◽  
Simulation Error ◽  
Spatially Varying

Efficient modeling schemes currently exist to handle the spatially variable-order fractional Laplacians in the fractional Laplacian viscoacoustic wave equation. The simplest approach is to change the spatially variable-order fractional Laplacians into a linear combination of several constant fractional-order Laplacians. We generalize the constant fractional-order scheme to a spatially variable fractional-order viscoelastic wave equation and develop an almost-equivalent constant fractional-order viscoelastic wave equation. Our constant fractional-order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order; thus, our scheme simulates seismic wave propagation in viscoelastic media with sharp [Formula: see text] contrasts well. The fast Fourier transform is used in the approximation of the fractional Laplacians, which improves the spectral accuracy in space. Several simulation examples are performed to verify that the numerical solution of a homogeneous [Formula: see text] model obtained by solving our constant fractional-order viscoelastic wave equation agrees well with that obtained by solving the original viscoelastic wave equation. The numerical simulations for spatially varying [Formula: see text] models obtained by the new wave equation are more straightforward than those currently in use and match the reference solutions obtained by accurate, but inefficient, methods. This match of simulation results verifies the accuracy of our viscoelastic wave equation.

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

2021 ◽  
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Jidong Yang ◽  
Zhenchun Li ◽  
Mukiibi Ssewannyaga Ivan
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Frequency Domain ◽  
Quality Factor ◽  
Fractional Order ◽  
Time Domain ◽  
Variable Order ◽  
Viscoelastic Wave Equation ◽  
Viscoelastic Wave ◽  
New Time

ABSTRACT Time-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

scholarly journals An analytical solution for solving a new wave equation within Lorenzo-Hartley kernel

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3739-3744
Author(s):  
Feng Gao
Keyword(s):  
Wave Equation ◽  
Analytical Solution ◽  
Fractional Order ◽  
New Wave ◽  
Liouville Type ◽  
Derivatives Of

In this article we investigate the general fractional-order derivatives of the Riemann-Liouville type via Lorenzo-Hartley kernel, general fractional-order integrals and the new general fractional-order wave equation defined on the definite domain with the analytical soluton.

Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Guangchi Xing ◽  
Tieyuan Zhu
Keyword(s):  
Wave Equation ◽  
Fractional Laplacian ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Physical Processes ◽  
Full Waveform ◽  
Adjoint State ◽  
The Finite Difference Method ◽  
Adjoint State Method ◽  
Laplacian Operators

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion ( Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. We find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel. After validating the Fréchet kernels using the finite-difference method, we conduct a numerical example to demonstrate the capability of the kernels to characterize both velocity and attenuation anomalies. The kernels of different misfit measurements are presented to investigate their different sensitivities. Our results suggest that rather than the traveltime, the amplitude and the waveform kernels are more suitable to capture attenuation anomalies. These kernels lay the foundation for the multiparameter inversion with the fractional formulation, and the decoupled nature of them promotes our understanding of the significance of different physical processes in the Q-FWI.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T121-T131 ◽  
Author(s):  
Tieyuan Zhu ◽  
Tong Bai
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Seismic Attenuation ◽  
Fourier Pseudospectral Method ◽  
Frequency Independent ◽  
Theoretical Predictions

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

scholarly journals Variable-Order Fractional Models for Wall-Bounded Turbulent Flows

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 782
Author(s):  
Fangying Song ◽  
George Em Karniadakis
Keyword(s):  
Fractional Derivative ◽  
Turbulent Flows ◽  
Fractional Order ◽  
Variable Order ◽  
Universal Curve ◽  
Reynolds Stresses ◽  
Continuous Change ◽  
Mean Velocity ◽  
Symmetric Variable ◽  
Fractional Models

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

scholarly journals A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aziz Khan ◽  
Hashim M. Alshehri ◽  
J. F. Gómez-Aguilar ◽  
Zareen A. Khan ◽  
G. Fernández-Anaya
Keyword(s):  
Fractional Order ◽  
Existence And Uniqueness ◽  
Variable Order ◽  
Fractional Order Systems ◽  
New Approach ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractThis paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.

scholarly journals Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

2021 ◽  
Vol 54 (1) ◽  
pp. 245-258
Author(s):  
Younes Bidi ◽  
Abderrahmane Beniani ◽  
Khaled Zennir ◽  
Ahmed Himadan
Keyword(s):  
Wave Equation ◽  
Global Solution ◽  
Fractional Laplacian ◽  
Necessary Conditions ◽  
Dynamic Structure ◽  
Decay Estimates ◽  
Damped Wave Equation ◽  
Structure Of Solutions ◽  
Nonlinear Source ◽  
Galerkin Approximations

Abstract We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.

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